Pulse flour in a food processor to aerate. Add eggs and process until dough forms a rough ball, about 30 seconds. (If dough resembles small pebbles, add water, 1/2 teaspoon at a time; if dough sticks to side of bowl, add flour, 1 tablespoon at a time, and process until dough forms a rough ball.)

Turn out dough and any small bits onto the counter or a silpat and knead by hand until dough is smooth, 1 to 2 minutes. Cover with plastic wrap and set aside to relax for at least 15 minutes or up to 2 hours.

Microwave the squash in a covered bowl until tender and easily pierced with a fork, 10 to 15 minutes. Drain squash well and transfer to a food processor. Add butter, brown sugar, 1/4 tsp salt, and nutmeg, and process until smooth, 15 to 20 seconds (you could use a blender if you don't have a food processor.) Transfer to a bowl, stir in Parmesan and 1/8 tsp pepper, and refrigerate until no longer warm, 15 to 25 minutes.

Dust 2 large rimmed baking sheets liberally with flour. Divide pasta dough into 5 even pieces and cover with plastic wrap. Working with 1 piece of dough, use a manual pasta machine to roll out dough into sheets. Flatten dough into a disk and run it through the rollers of the pasta machine set to the widest position. Roll pasta thinner by putting it through the machine repeatedly, narrowing the setting each time. Roll until the dough is thin and satiny and you can see the outline of your hand through the pasta. Lay the pasta sheet over a floured ravioli maker (it will need to double back over the top after you fill the ravioli.) Place generous 1-teaspoon dollops of filling in the center of each ravioli. Fold the remaining pasta over the filling and use a rolling pin, roll firmly across the ravioli tray to seal the ravioli. 

Alternatively without a ravioli maker, using a pizza wheel or sharp knife, cut pasta sheet into long rectangles measuring 4 inches across. Place generous 1-teaspoon dollops of filling over bottom half of dough, spaced about 1 1/4 inches apart. (If dough seems dry, dab with water.) Fold top of pasta over filling and press layers of dough securely around each filling to seal. Use fluted pastry wheel to cut ravioli apart and trim edges (or just a knife or pizza cutter will do the trick, and you will have straight edges rather than zig-zags).

Transfer finished ravioli to the floured baking sheet and cover with a clean, damp dish towel. Repeat with remaining pasta and filling. (Towel-covered baking sheets of ravioli can be wrapped in plastic wrap and refrigerated for up to 4 hours. Ravioli can also be frozen for up to 1 month. Place the whole baking sheet into the freezer for a few hours to flash freeze the ravioli, then you can transfer them to a ziplock freezer bag. When you cook the ravioli, do not thaw before boiling.)

Before cooking the ravioli, cook butter, hazelnuts, sage and 1/4 tsp salt in a 10-inch skillet over medium-high heat, swirling pan constantly, until butter is melted, has golden brown color/a few specks of brown, and releases a nutty aroma, about 3 minutes. Off heat, stir in lemon juice.

Bring 4 quarts water to a boil in a large pot (I begin heating the water as I make the sauce to save on lag time while waiting for it to boil.) Add 1 tablespoon salt and half of the ravioli. Cook, stirring often and lowering heat if necessary to keep water at a gentle boil, until ravioli are tender, about 2 minutes (3 to 4 minutes if frozen). Using a slotted spoon, transfer cooked ravioli to a serving platter, spoon some of the butter sauce over top, and cover with aluminum foil to keep warm. Return water to a boil and repeat to cook remaining ravioli. Swirl 2 tablespoons of ravioli cooking water into remaining butter sauce, then pour sauce over ravioli. Top with shaved Parmesan and serve immediately.